# On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne

About: This article is published in Annals of Mathematics.The article was published on 1992-05-01. It has received 59 citations till now. The article focuses on the topics: Conjecture.

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TL;DR: In this paper, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field, which generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves.

Abstract: A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$ We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter
The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here
Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations

147 citations

### Cites background from "On the calculation of local terms i..."

...When X and S descend to a finite field, the projection S → X is proper, and S → X ′ is quasi-finite, Theorem 1.1 follows from Deligne’s conjecture ([Fujiwara97],[Pink92]) together with his theorem on eigenvalues of Frobenius....

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...In general these last two assumptions ([Pink92] 7.1.1) cannot be simultaneously obtained in our context, as far as I can see....

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TL;DR: The Lefschetz-Verdier trace formula as discussed by the authors expresses the global trace of an endomorphism induced on cohomology by a correspondence as a sum of local contributions, called local terms, near fixed points.

Abstract: In abstract algebraic geometry, the Lefschetz-Verdier trace formula expresses the global trace of an endomorphism induced on cohomology by a correspondence as a sum of local contributions, called local terms, near fixed points. Though the formula holds quite generally, the explicit calculation of local terms is very hard to carry out, and no good formula is known for general correspondences. Over the complex numbers a useful formula is known for a good class called weakly hyperbolic [G-M]. On the other hand, in positive characteristics, P. Deligne conjectured that the local terms would be much simplified if we composed a given correspondence with a high power of Frobenius. Take a proper variety X over an algebraic closure k of a finite field Fq, a correspondence a : Y ~ X xk X, and a smooth Qr K on an open set U of X fixed by a (denote the inclusion by j , and { is invertible in k). Then

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21 Feb 2007TL;DR: In this paper, the authors generalize a theorem of Fujiwara (Deligne's conjecture) to the situation appearing in a joint work with Kazhdan on the global Langlands correspondence over function fields.

Abstract: The goal of this paper is to generalize a theorem of Fujiwara (Deligne’s conjecture) to the situation appearing in a joint work [KV] with David Kazhdan on the global Langlands correspondence over function fields. Moreover, our proof is more elementary than the original one and stays in the realm of ordinary algebraic geometry, that is, does not use rigid geometry. We also give a proof of the Lefschetz–Verdier trace formula and of the additivity of filtered trace maps, thus making the paper essentially self-contained.

57 citations

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2,896 citations

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TL;DR: In this article, the authors implique l'accord avec les conditions generales d'utilisation (http://www.numdam.org/legal.php).

Abstract: © Publications mathematiques de l’I.H.E.S., 1974, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

2,792 citations

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Abstract: JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics.

798 citations

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TL;DR: In this paper, a prime number is defined as the completion of the algebraic closure of the field of rational p-adic numbers and A is the residue class field of Q. The non-archimedean valuation of 0 will be denoted by the ordinal function, abbreviated "ord ", and normalized by the condition ord p = 1.

Abstract: Let p be a prime number, a2 the completion of the algebraic closure of the field of rational p-adic numbers and let A be the residue class field of Q. The field A is the algebraic closure of its prime subfield and is of characteristic p. If T* is the set of all roots of unity in a2 of order prime to p then the restriction of the residue class map to T* is a multiplicative isomorphism of T* onto the multiplicative group of R. The elements of T T* U {O} form the Teichmiiller representatives of A in Q2 and for each x C A the representative of x in Q will be understood to be the element of T in the class x. The non-archimedean valuation of 0 will be denoted by the ordinal function, abbreviated "ord ", and normalized by the condition ord p = 1.

397 citations